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Proof step 42 from proof max_exp-fac-gen (previous step)

max_exp(i, fac(i1)) = Σ countFun(1, i0 - 1, λ j1. i1 / i ^ j1), (i1 * i + j2) / i = i1, 0 < i1, 0 < i1 * i + j2, 0 < i, 0 ≤ j2, prime(i), i1 * i + j2 < i ^ i0, 0 < i ^ i0, 0 ≤ i0, 1 < i ^ i0, j2 < i, 1 < i0, 0 < i0, ∀ j0, i2, i0. i→n(j0) < i→n(i1 * i + j2) → prime(i2) ∧ j0 < i2 ^ i0 ∧ 0 < j0 ∧ 0 < i2 ^ i0 ∧ 0 ≤ i0 ∧ 1 < i2 ^ i0 → max_exp(i2, fac(j0)) = Σ countFun(1, i0, λ j1. j0 / i2 ^ j1) ⊦ max_exp(i, fac(i1)) = Σ countFun(2, i0, λ j0. (i1 * i + j2) / i ^ j0), i→n(i0) - 1 = # countFun(2, i0, λ j0. (i1 * i + j2) / i ^ j0) ∧ (n + 1 < i→n(i0) → countFun(1, i0 - 1, λ j1. i1 / i ^ j1)[n] = countFun(2, i0, λ j0. (i1 * i + j2) / i ^ j0)[n])

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: intlist inst: name: countFun-length seq: ⊦ 0 ≤ i ∧ i ≤ j → # countFun(i, j, intfun) = i→n(1 + j - i) subst: [intfun, j, i] → [λ j0. (i1 * i + j2) / i ^ j0, i0, 2] flags: rewrite

Additional information

Used simplifier rules: lels-01: ⊦ ¬ i ≤ j ↔ j < i; from specification int-abs%two-ls: ⊦ 0 < j → (j < 2 ↔ j = 1); from specification int-basic1%sub-eq-01: ⊦ i = j → i - j = 0; from specification int-abs%lel: ⊦ i < i0 ∧ i = i1 → i1 < i0; (forward) from specification oelem[intlista]%irreflexivity: ⊦ ¬ i < i; from specification oelem[intlista]%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic1%l: ⊦ # ints = 0 ↔ ints = []; from specification list-data[intlista] Used terms: λ j0. (i1 * i + j2) / i ^ j0, i0, 2